Pith. sign in

REVIEW 1 major objections 7 minor 33 references

Brockett cost function for symplectic eigenvalues

T0 review · 1 major / 7 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper introduces a weighted trace cost function of Brockett type for symplectic eigenvalues and proves that, when the weight matrix has strictly increasing positive entries, every critical point is exactly a symplectic eigenvector…

desk verdict Correct and clean paper on a Brockett cost for symplectic eigenpairs; the main theorems hold, but the motivation for distinct weights overreaches Lemma 2.2. read the letter →

arxiv 2506.07560 v1 pith:6CTFDNE5 submitted 2025-06-09 math.OC

Nguyen Thanh Son This is my paper
classification math.OC MSC 15A1515A1870G45
keywords BrockettcostfunctionsymplecticeigenvaluesWilliamson'sdiagonalformtraceminimizationStiefelmanifoldcriticalpointsrearrangementinequalitysymmetricpositive-definitematrices
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Symplectic eigenvalues are the positive numbers in Williamson's normal form of a symmetric positive-definite matrix under symplectic congruence; they appear in quantum information, optics, and Hamiltonian systems. The paper shows that a weighted trace function of Brockett type, minimized over symplectic matrices with distinct positive weights, has the property that every critical point is a genuine symplectic eigenvector set. This means one optimization step recovers both the eigenvalues and the eigenvectors, whereas the classical trace-minimization approach only yielded a basis for the eigenspace. The paper also derives the exact global minimum, identifies saddle points, and recovers the earlier trace-minimization theorem as a limiting case.

What carries the argument

The carrying object is the Brockett cost function $f(X)=\operatorname{tr}(\tilde{N} X^T M X)$ on the symplectic Stiefel manifold $Sp(2k,2n)$, with $\tilde{N}=\operatorname{diag}(N,N)$ and $N$ having pairwise distinct positive entries. The argument runs through the first-order condition $M X\tilde{N} = J_{2n}XL$ for a skew-symmetric Lagrange multiplier $L$; combining this with the symmetry of $X^T M X$ and the distinctness of the weights invokes Lemma 2.2(ii) (a matrix commuting with a diagonal matrix of distinct entries is diagonal) to conclude $X^T M X = \operatorname{diag}(S_1,S_1)$. The auxiliary group $OrSp_{\tilde{N}}(2k)$ of symplectic matrices that also preserve $\tilde{N}$, explicitly block rotations $\begin{pmatrix}\operatorname{diag}(\cos\Phi)&\operatorname{diag}(\sin\Phi)\\-\operatorname{diag}(\sin\Phi)&\operatorname{diag}(\cos\Phi)\end{pmatrix}$, gives the cost its invariance and transports critical points to critical points.

What would settle it

Take $M = \operatorname{diag}(1,2,3,4)$ and $k = n = 2$ with equal weights $N = \operatorname{diag}(1,1)$ in (3.6), and solve the critical-point equations; if a critical point $X$ has $X^T M X$ with nonzero off-diagonal blocks or unequal diagonal blocks, the distinctness assumption is doing essential work and Theorem 3.4 cannot hold for repeated weights.

Watch

Extended reading notes

Core claim

The central result (Theorem 3.4) is an equivalence: for $M \in SPD(2n)$ and $\tilde{N} = \operatorname{diag}(N,N)$ with $N = \operatorname{diag}(\nu_1,\ldots,\nu_k)$, $0<\nu_1<\cdots<\nu_k$, a matrix $X \in \mathbb{R}^{2n\times 2k}$ satisfying $X^T J_{2n}X = J_{2k}$ is a critical point of $f(X)=\operatorname{tr}(\tilde{N} X^T M X)$ if and only if its columns form a symplectic eigenvector set of $M$, i.e. $X^T M X = \operatorname{diag}(D,D)$ with $D$ diagonal and $M X = J_{2n} X \begin{pmatrix}0&-D\\D&0\end{pmatrix}$. The proof forces $X^T M X$ to be block-scalar through the distinctness of the weights; conversely, every symplectic eigenvector set satisfies the first-order condition with an explicit Lagrange multiplier. Theorem 3.5 gives the global minimum as $2(d_1\nu_k+\cdots+d_k\nu_1)$, attained by eigenvectors paired with the smallest $k$ eigenvalues in reverse order, and Remark 3.7 shows the classical trace-minimization theorem follows by taking $N=\operatorname{diag}(1+\epsilon,\ldots,1+k\epsilon)$ and letting $\epsilon \downarrow 0$. Proposition 3.8 adds that critical points with eigenvalues not in nonincreasing order are saddle points.

Load-bearing premise

The weight matrix $N$ must have strictly increasing entries, with no two equal; if two weights coincide, the step that forces the off-diagonal blocks of $X^T M X$ to vanish no longer works, and the equivalence between critical points and symplectic eigenvector sets can break down.

Editorial extensions

If this is right

  • A single Riemannian optimization run over the symplectic Stiefel manifold with distinct weights yields the symplectic eigenpairs directly, with no second diagonalization step.
  • The global minimum formula orders the output: the minimizer pairs the smallest symplectic eigenvalue with the largest weight, so the returned columns are automatically sorted in nonincreasing eigenvalue order.
  • The trace-minimization theorem of Theorem 2.1 is a corollary: let the weights approach equality in the Brockett function and pass to the limit.
  • Any critical point that is not a minimizer is a saddle point, so gradient-type methods that avoid saddles will converge to genuine eigenvector sets.
  • The critical-point equivalence (Theorem 3.4) and the saddle-point statement remain valid for positive-semidefinite and real symmetric matrices, as noted in the concluding remarks; only the global minimum formula and its corollaries need positive definiteness.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • By analogy with the classical Brockett flow for sorting eigenvalues, the weighted symplectic trace likely generates a gradient flow on the symplectic Stiefel manifold that sorts symplectic eigenvalues; this would give a dynamical-system algorithm that outputs eigenpairs continuously in time.
  • Modding out by $OrSp_{\tilde{N}}(2k)$ collapses the symmetry and may turn the cost into a Morse-Bott function on a smaller quotient manifold, which could simplify convergence analysis for Riemannian optimizers.
  • The paper leaves open whether a change of variables can bring Theorem 3.5 under the general Brockett principle for matrix pencils from [24]; if such a bridge exists, it would extend the result to indefinite or singular $B$ and cover structured Hamiltonian eigenvalue problems.
  • A practical test: certify a computed critical point as a saddle or minimizer by checking the sign of the Hessian expression in Proposition 3.8, which needs only the diagonal of $X^T M X$ and the weights.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 7 minor

Summary. The paper introduces a Brockett-type cost function f(X) = tr(\tilde{N} X^T M X) for a symmetric positive-definite matrix M, subject to the symplectic constraint X^T J_{2n} X = J_{2k}, where \tilde{N} = diag(N,N) with N = diag(\nu_1,\dots,\nu_k) and distinct positive weights. The main results are Theorem 3.4, which characterizes critical points as symplectic eigenvector sets; Theorem 3.5, which gives the global minimum value as 2(d_1\nu_k + \cdots + d_k\nu_1) and describes the minimizers; and Proposition 3.8, which provides a sufficient condition for a critical point to be a saddle point. Remark 3.7 derives the known trace minimization theorem for symplectic eigenvalues as a limiting case. The proofs use standard tools including Lagrange multiplier conditions, Lemma 2.2, and the rearrangement inequality.

Significance. If correct, this is a valuable contribution to the theory of computing symplectic eigenpairs via optimization. Unlike the prior trace minimization approach, which yields only a subspace basis, the Brockett cost function allows critical points to directly provide symplectic eigenvectors. The derivation of the classical trace minimization theorem as a consequence of the new result is elegant and demonstrates the strength of the framework. The proofs are self-contained and checkable, with the only free parameter being the weight matrix N with distinct positive entries, whose role is transparent. The paper should be of interest to researchers in Riemannian optimization, Hamiltonian eigenvalue problems, and quantum information. The main limitation is the necessity of distinct weights, which the paper currently presents as derived rather than as a standing assumption, and this needs clarification.

major comments (1)
  1. [Section 3, Eq. (3.5)] The assertion that Lemma 2.2(i) forces \tilde{N} to take the form (3.5) with mutually distinct diagonal entries is not supported. Lemma 2.2(i) only shows that the entries of \tilde{N}^{-1} pair up as r_j = r_{j+k}, i.e., \tilde{N} = diag(N,N); it does not imply \nu_i \neq \nu_j. The distinctness is an additional hypothesis, and it is in fact necessary for the equivalence in Theorem 3.4: for M = diag(1,2,1,2), k = 2, \tilde{N} = I_4, and X = diag(Q,Q) with Q a non-diagonal orthogonal matrix, one checks that X is symplectic and satisfies the first-order condition (3.2) with L = J_4^T X^T M X, yet X^T M X is not diagonal and X is not a symplectic eigenvector set. I recommend rewriting the passage around (3.5) to state that distinctness is imposed as a standing assumption, and adding a remark that the equivalence genuinely depends on it. The proofs of Theorems 3.4 and 3.5 are otherwise correct under the stated hypothesis.
minor comments (7)
  1. [Theorem 3.5] The formula "X_*^T M X_* = diag(d_k,\dots,d_1)" is dimensionally imprecise: the left-hand side is a 2k by 2k matrix, so the right-hand side should be diag(D,D) with D = diag(d_k,\dots,d_1).
  2. [Remark 3.7] The continuity argument that takes the limit \epsilon \to 0 in (3.15) is terse; it would be helpful to justify that the minimizers of the weighted problems stay in a common bounded set (e.g., using the coercivity bound in (3.14)), so that the limit can be interchanged with the minimization.
  3. [Abstract and Section 1] The phrase "any critical point consists of symplectic eigenvectors" is imprecise; each symplectic eigenvector is a pair of vectors, so it is more accurate to say "is a symplectic eigenvector set" or "consists of symplectic eigenvector pairs."
  4. [Proof of Theorem 3.4] In the sentence "we obtain the quality", "quality" should be "equality".
  5. [Page 1] "It is worth to note" should be "It is worth noting".
  6. [Remark 3.6] "c.f." should be "cf.".
  7. [Concluding remarks] "optimization respects" should be "optimization aspects".

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: The Brockett-critical-point equivalence and the global minimum are proved from first-order conditions and the rearrangement inequality; self-citations are contextual only, with a minor overstatement at (3.5).

full rationale

The derivation chain is self-contained. The Brockett cost function is constructed at (3.6) with Ñ=diag(N,N) and N having distinct positive entries as a deliberate design choice; no parameter is fitted to the target symplectic data. Theorem 3.4 is proved from the first-order condition (3.2): the key step is that (3.10) forces X^T M X = diag(S1,S1) using Lemma 2.2(ii), and substituting back gives the symplectic eigenvector relation; the converse is a direct substitution. Theorem 3.5 then uses coercivity, Weierstrass's theorem, and the classical rearrangement inequality, and it does not invoke Theorem 2.1 as an input. Remark 3.7's recovery of the trace-minimization theorem is a limit argument letting epsilon tend to zero in the formula of Theorem 3.5, so it is a corollary rather than an input. Citations [18,7] state Theorem 2.1 as background but that theorem is not used in the proofs of Theorems 3.4 and 3.5; the self-citations [29] and [14] are used only for context or standard facts such as the basis property of trace minimizers and LICQ, so they are not load-bearing for the main equivalence. The only questionable sentence is around (3.5): Lemma 2.2(i) forces the block-diagonal form Ñ=diag(N,N) but does not by itself force the entries of N to be mutually distinct, even though the passage says Ñ 'must take the form (3.5)' including distinctness. This is an expositional overstatement rather than circularity, because Theorem 3.4 explicitly assumes 0<ν1<...<νk and the proof uses that assumption. No fitted parameter is renamed as a prediction, and the main results do not reduce to their assumptions by construction.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted to data; the weight matrix N is a user-chosen parameter with a structural distinctness condition. The proofs rely on standard theorems in matrix analysis and optimization as listed. No new physical or mathematical entities are postulated.

free parameters (1)
  • weight matrix N (diagonal entries nu_1,...,nu_k), distinct and positive
    The Brockett cost function (3.6) is parameterized by N = diag(nu_1,...,nu_k). The theorems hold for any N with 0 < nu_1 < ... < nu_k; no numeric values are fitted to data. The distinctness condition is load-bearing for Theorem 3.4.
assumptions (6)
  • standard math Williamson's theorem: every spd matrix M in R^(2n x 2n) admits a symplectic matrix S with S^T M S = diag(D,D).
    Invoked in the introduction to define symplectic eigenvalues and used as background throughout.
  • standard math Lagrange multiplier first- and second-order necessary optimality conditions for smooth equality-constrained optimization.
    Used in Section 3 to derive the critical point equation (3.2) and second-order condition (3.3).
  • standard math Weierstrass extreme value theorem: a continuous coercive function on a closed set has a minimizer.
    Used in the proof of Theorem 3.5 to show existence of a global minimizer on the closed unbounded symplectic Stiefel set.
  • standard math Rearrangement inequality: for sorted sequences, the sum of products is minimized by pairing opposite orders.
    Used in Theorem 3.5 and Proposition 3.8 to identify the minimal critical value and saddle points.
  • domain assumption The feasible set Sp(2k,2n) is closed and unbounded in R^(2n x 2k).
    Relied on in the coercivity argument in Theorem 3.5; closedness follows from the polynomial constraint, unboundedness is evident.
  • domain assumption The target matrix M is symmetric positive-definite.
    The main theorems are stated and proved only for this class; the PSD case is mentioned in remarks but not covered by Theorem 3.5.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Brockett cost function for symplectic eigenvalues." pith.science (2026). https://pith.science/paper/6CTFDNE5

@misc{pith2026250607560,
  author       = {Pith},
  title        = {Pith review of: Brockett cost function for symplectic eigenvalues},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6CTFDNE5}},
  note         = {Machine review of arXiv:2506.07560}
}
read the original abstract

The symplectic eigenvalues and corresponding eigenvectors of symmetric positive-definite matrices in the sense of Williamson's theorem can be computed via minimization of a trace cost function under the symplecticity constraint. The optimal solution to this problem only offers a symplectic basis for a symplectic eigenspace corresponding to the sought symplectic eigenvalues. In this paper, we introduce a Brockett cost function and investigate the connection between its properties and the symplectic eigenvalues and eigenvectors, specifically prove that any critical point consists of symplectic eigenvectors. Surprisingly, the trace minimization theorem for the symplectic eigenvalues can be deduced from our results.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

33 extracted references · 19 canonical work pages

  1. [1]

    Absil, R

    P.-A. Absil, R. Mahony, and R. Sepulchre , Optimization Algorithms on Matrix Manifolds , Princeton University Press, Princeton, NJ, 2008

    2008

  2. [2]

    Adesso and F

    G. Adesso and F. Illuminati, Entanglement in continuous-variable systems: recent advances and current perspectives, J. Phys. A: Math. Theor., 40 (2007), p. 7821, https://doi.org/10. 1088/1751-8113/40/28/S01

    work page 2007

  3. [3]

    Amodio, On the computation of few eigenvalues of positive definite Hamiltonian matrices , Future Gener

    P. Amodio, On the computation of few eigenvalues of positive definite Hamiltonian matrices , Future Gener. Comput. Syst., 22 (2006), pp. 403–411, https://doi.org/10.1016/j.future. 2004.11.027

    doi:10.1016/j.future 2006

  4. [4]

    Banchi, S

    L. Banchi, S. L. Braunstein, and S. Pirandola , Quantum fidelity for arbitrary gaussian states, Phys. Rev. Lett., 115 (2015), p. 260501, https://doi.org/10.1103/PhysRevLett.115. 260501

    doi:10.1103/physrevlett.115 2015

  5. [5]

    Benner, D

    P. Benner, D. Kressner, and V. Mehrmann , Skew-Hamiltonian and Hamiltonian eigen- value problems: Theory, algorithms and applications , in Proceedings of the Conference on Applied Mathematics and Scientific Computing, 2005, pp. 3–39, https://doi.org/10.1007/ 1-4020-3197-1 1

    work page 2005

  6. [6]

    D. P. Bertsekas, Nonlinear Programming, Athena Scientific, 2nd ed., 1999. 10 NGUYEN THANH SON

    work page 1999

  7. [7]

    Bhatia and T

    R. Bhatia and T. Jain , On the symplectic eigenvalues of positive definite matrices , J. Math. Phys., 56 (2015), p. 112201, https://doi.org/10.1063/1.4935852

    doi:10.1063/1.4935852 2015

  8. [8]

    Birtea, I

    P. Birtea, I. Cas ¸u, and D. Com˘anescu, First order optimality conditions and steepest descent algorithm on orthogonal stiefel manifolds , Optim. Lett., 13 (2019), pp. 1773–1791, https: //doi.org/10.1007/s11590-018-1319-x

    work page doi:10.1007/s11590-018-1319-x 2019

Show all 33 references
  1. [9]

    Brockett, Least squares matching problems, Linear Algebra Appl., 122-124 (1989), pp

    R. Brockett, Least squares matching problems, Linear Algebra Appl., 122-124 (1989), pp. 761– 777, https://doi.org/10.1016/0024-3795(89)90675-7. Special Issue on Linear Systems and Control

    1989 doi

  2. [10]

    Brockett, Dynamical systems that sort lists, diagonalize matrices, and solve linear pro- gramming problems, Linear Algebra Appl., 146 (1991), pp

    R. Brockett, Dynamical systems that sort lists, diagonalize matrices, and solve linear pro- gramming problems, Linear Algebra Appl., 146 (1991), pp. 79–91, https://doi.org/10.1016/ 0024-3795(91)90021-N

    1991

  3. [11]

    Chakraborty, A

    M. Chakraborty, A. Mukherjee, I. Krikidis, A. Nag, and S. Chandra, A hybrid noise ap- proach to modelling of free-space satellite quantum communication channel for continuous- variable qkd , IEEE Trans. Green Commun. Netw., (2025), pp. 1–1, https://doi.org/10. 1109/TGCN.2024.3525297

    2025

  4. [12]

    de Gosson, Symplectic Geometry and Quantum Mechanics, Advances in Partial Differential Equations, Birkh¨ auser, Basel, 2006

    M. de Gosson, Symplectic Geometry and Quantum Mechanics, Advances in Partial Differential Equations, Birkh¨ auser, Basel, 2006

    2006

  5. [13]

    Eisert, T

    J. Eisert, T. Tyc, T. Rudolph, and B. Sanders, Gaussian quantum marginal problem, Com- mun. Math. Phys., 280 (2008), pp. 263–280, https://doi.org/10.1007/s00220-008-0442-4

    2008 doi

  6. [14]

    B. Gao, N. T. Son, P.-A. Absil, and T. Stykel , Riemannian optimization on the symplectic Stiefel manifold , SIAM J. Optim., 31 (2021), pp. 1546–1575, https://doi.org/10.1137/ 20M1348522

    2021

  7. [15]

    B. Gao, N. T. Son, and T. Stykel , Symplectic Stiefel manifold: tractable metrics, second- order geometry and Newton ’s methods , arxiv preprint, 2024, https://doi.org/10.48550/ arXiv.2406.14299

    2024 doi arXiv

  8. [16]

    Golub and C

    G. Golub and C. V. Loan, Matrix Computations. 4th ed, The Johns Hopkins University Press, Baltimore, London, 2013

    2013

  9. [17]

    Hardy, J

    G. Hardy, J. E. Littlewood, and G. P ´olya, Intequalities, Cambridge University Press, London, 1934

    1934

  10. [18]

    Hiroshima, Additivity and multiplicativity properties of some Gaussian channels for Gauss- ian inputs , Phys

    T. Hiroshima, Additivity and multiplicativity properties of some Gaussian channels for Gauss- ian inputs , Phys. Rev. A, 73 (2006), p. 012330, https://doi.org/10.1103/PhysRevA.73. 012330

    2006 doi

  11. [19]

    Jain and H

    T. Jain and H. K. Mishra , Derivatives of symplectic eigenvalues and a Lidskii type theorem , Canad. J. Math., 74 (2022), pp. 457–485, https://doi.org/10.4153/S0008414X2000084X

    2022 doi

  12. [20]

    Jensen and R

    R. Jensen and R. Zimmermann , Riemannian optimization on the symplectic Stiefel manifold using second-order information, 2024, https://doi.org/10.48550/arXiv.2404.08463

    2024 doi arXiv

  13. [21]

    Kovaˇc-Striko and K

    J. Kovaˇc-Striko and K. Veseli ´c, Trace minimization and definiteness of symmetric pen- cils, Linear Algebra Appl., 216 (1995), pp. 139–158, https://doi.org/10.1016/0024-3795(93) 00126-K

    1995 doi

  14. [22]

    Krbek, T

    M. Krbek, T. Tyc, and J. Vlach, Inequalities for quantum marginal problems with continuous variables, J. Math. Phys., 55 (2014), p. 062201, https://doi.org/10.1063/1.4880198

    2014 doi

  15. [23]

    Liang and R.-C

    X. Liang and R.-C. Li , On generalizing trace minimization principles, II , Linear Algebra Appl., 687 (2024), pp. 8–37, https://doi.org/10.1016/j.laa.2024.01.015

    2024 doi

  16. [24]

    Liang, L

    X. Liang, L. W ang, L.-H. Zhang, and R.-C. Li , On generalizing trace minimization prin- ciples, Linear Algebra Appl., 656 (2023), pp. 483–509, https://doi.org/10.1016/j.laa.2022. 10.012

    2023 doi

  17. [25]

    H. Liu, A. So, and W. Wu , Quadratic optimization with orthogonality constraint: explicit Lojasiewicz exponent and linear convergence of retraction-based line-search and stochastic variance-reduced gradient methods, Math. Program. Ser. A, 178 (2019), pp. 215–262, https: //doi....

    2019 doi

  18. [26]

    C. V. Loan, A symplectic method for approximating all the eigenvalues of a Hamiltonian ma- trix, Linear Algebra Appl., 61 (1984), pp. 233–251, https://doi.org/10.1016/0024-3795(84) 90034-X

    1984 doi

  19. [27]

    H. K. Mishra , On generalization of Williamson ’s theorem to real symmetric matrices , 2024, https://doi.org/10.48550/arXiv.2408.04894

    2024 doi arXiv

  20. [28]

    Ortega and D

    J.-P. Ortega and D. Yin , Learnability of linear port-hamiltonian systems , J. Mach. Learn. Res., 25 (2024), pp. 1–56, http://jmlr.org/papers/v25/23-0450.html

    2024

  21. [29]

    N. T. Son, P.-A. Absil, B. Gao, and T. Stykel, Computing symplectic eigenpairs of symmet- ric positive-definite matrices via trace minimization and Riemannian optimization. , SIAM J. Matrix Anal. Appl., 42 (2021), pp. 1732–1757, https://doi.org/10.1137/21M1390621

    2021 doi

  22. [30]

    N. T. Son and T. Stykel , Symplectic eigenvalues of positive-semidefinite matrices and the trace minimization theorem , Electron. J. Linear Algebra, 38 (2022), pp. 607–616, https: BROCKETT COST FUNCTION FOR SYMPLECTIC EIGENV ALUES 11 //doi.org/10.13001/ela.2022.7351

    2022

  23. [31]

    Weedbrook, S

    C. Weedbrook, S. Pirandola, R. Garcia-Patron, N. Cerf, T. Ralph, J. Shapiro, and S. Lloyd, Gaussian quantum information, Rev. Mod. Phys., 84 (2012), pp. 621–669, https: //doi.org/10.1103/RevModPhys.84.621

    2012 doi

  24. [32]

    Williamson , On the algebraic problem concerning the normal forms of linear dynamical systems, Am

    J. Williamson , On the algebraic problem concerning the normal forms of linear dynamical systems, Am. J. Math., 58 (1936), pp. 141–163

    1936

  25. [33]

    Yamada and H

    M. Yamada and H. Sato, Conjugate gradient methods for optimization problems on symplectic Stiefel manifold , IEEE Control Syst. Lett., 7 (2023), pp. 2719–2724, https://doi.org/10. 1109/LCSYS.2023.3288229

    2023

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.